The Commodore is a member of the Captain's Council, and is first among equals. The Commodore has the ultimate authority in Lion's Arch, however, he rarely makes a decision or move without the support of the Council. While the Commodore can't overrule the council on any one issue, his vote means more than that of the other captains, especially when appointing new council members.
At a given time, an Order of Whispers agent is kept aware of the happenings of the council, by agreement with the Commodore. In exchange, the Order keeps the Commodore aware of thieves, smuggling rings, and other dangers.
The first Commodore was Captain Cobiah Marriner, who sat on the first Captain's Council. The original council was politically egalitarian, and the title of "commodore" was just an affectionate name for Captain Marriner given to him by the citizens of Lion's Arch. When Prince Edair formally recognised the independence of Lion's Arch from Kryta in 1256 AE, he did so with the requirements that Lion's Arch defend Kryta against the Dead Ships and be run by a Council rather than a single individual. This effectively codified the title.
The current Commodore is Lawson Marriner, grandson of Cobiah Marriner, who stepped into the role in 1300 AE.